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Here is a basic introduction to amateur telescopes. Celestron International also have an excellent article about telescope basics.
Types of Telescopes
There are basically two types of telescopes, the refractor telescope and the reflector telescope.
The refractor uses a front objective lens to collect light and focus it onto the eyepiece at the other end of the telescope. A second lens behind the main front lens is necessary to correct for false colours that would otherwise show up in the eyepiece.

The Newtonian reflector uses a mirror at the back end to collect and focus light onto a smaller secondary mirror which diverts the light into the eyepiece. The eyepiece is responsible for magnifying the image so it is visible to the human eye. There are several different types of reflector telescopes.

The Newytonian is described above. The Schmidt-Cassegrain reflector is like a combination of a reflector and refractor. A spherical mirror at the back end collects light and focuses it onto a small secondary at the front which then directs it back through a hole in the main primary mirror and into the eyepiece. The primary mirror in a SCT telescope produces severe aberrations so a corrector lens is placed at the front of the telescope to correct for these.
The Maksutov-Cassegrain is identical to the Schmidt Cassegrain but has a different front lens to again correct for aberrations.

Telescope Mounts
There are two types of mounts for telescopes:
1. The Alt/Azimuth Mount
2. The Equatorial Mount
The alt/azimuth mount moves the telescope only in altitude (up and down) and azimuth (right and left). It is not possible to accurately track star positions with this type of mount. Field rotation will occur when tracking for long periods. Most refractors and Schmidt-Cassegrain telescopes are fork mounted, which is a type of alt/azimuth mount. Large Newtomian reflectors are usually Dobsonian mounted, another type of alt/azimuth mount.

Dobsonian Mounted Newt
The equatorial mount moves the telescope in right ascension and declination. This mount can accurately track the position of a star if a clock drive motor is fitted to the right ascension axis.

Equatorial Mounted Reflector
Eyepieces
Plossl Eyepieces - The best type of eyepiece for amateur astronomy. A 4 element design that gives sharp images with a wide field of view and good eye relief.
Eyepieces are identified by focal length. Popular focal lengths are 10mm and 25mm. The shorter the focal length, the higher the magnification.
Magnification =
focal length of telescope
focal length of eyepiece
The highest effective magnifiaction in a telescope is 50 x aperture in inches.
Barlow Lens
This inserts between tthe focuser and eyepiece and doubles the magnification of the eyepieces.
Magnification
In astronomy, the resolution of the telescope, or light gathering ability, is more important than the magnification. Although a 60mm refractor might be capable of 400x magnification, the effective magnification is only 50 x 2.4inches = 120x as the lens does not have good light gathering ability.
The highest effective magnification in a telescope is 50 x aperture in inches.
i.e. 8inch (200mm) Newtonian = 50 x 8 = 400x
To calculate the actual magnification of a telescope, use the following formula:
Magnification =
focal length of telescope
focal length of eyepiece
i.e. 200mm f/6 with 10mm eyepiece
mag = (200 x 6)/10 = 120x
Focal Length
The focal length of a telescope is the distance the primary lens focuses light to a single point. Focal length is often expressed as an f/ratio:
f/ratio = focal length of lens/aperture of lens
i.e. 200mm lens with a focal length of 1200mm
f/ratio = 1200/200 = f/6
Collimation
Newtonian telescopes can easily be knocked out of alignment. When this happens, the mirrors need to be recentered so the optic axis coincides with the optic axis of the eyepiece. To do this, a collimating eyepiece is used.